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\begin{document}

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%   标题
\title{Amplitude-Sensitive Permutation Entropy: A Novel Complexity Measure Incorporating Amplitude Variation for Physiological Time Series} %Title of paper


%   作者信息
\author{Jun Huang}

\author{Huijuan Dong}

\author{Na Li}

\author{Yizhou Li}

\author{Jing Zhu}

\author{Xiaowei Li}
\altaffiliation{lixwei@lzu.edu.cn}


\author{Bin Hu}
\altaffiliation{bh@lzu.edu.cn}
\affiliation{School of Information Science and Engineering, Lanzhou University, No. 222 South Tian Shui Road, Lanzhou, 730000, Gansu, China}


\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

%   摘要
\begin{abstract}
% 事实证明，置换熵（PE）方法是对生理时间序列进行非线性分析的可靠工具，能为了解生物系统的基本动态提供宝贵的见解。作为生理数据的重要组成部分，振幅变化反映了神经活动、心血管动态和病理变化等关键过程。传统的幅值分析方法主要关注数据序列中的顺序模式，往往忽略了蕴含关键信息的幅值变化。本研究引入了一种新的复杂度测量方法，称为振幅敏感置换熵（ASPE）。ASPE 根据变异系数对序数模式进行加权，从而整合了振幅信息。ASPE 的性能首先通过模拟进行了评估，它在检测振幅变化和准确分配权重方面优于现有的五种 PE 方法。为了证实其更广泛的适用性，ASPE 随后在两个振幅丰富的生理时间序列数据集上进行了测试。结果表明，ASPE 不仅能更有效地捕捉动态变化，而且在区分疾病状态方面也超越了现有方法。
  Permutation entropy (PE) methods have proven to be robust tools for the nonlinear analysis of physiological time series, offering valuable insights into the underlying dynamics of biological systems. As a critical component of physiological data, amplitude variation reflects key processes such as neural activity, cardiovascular dynamics, and pathological changes. Traditional PE methods predominantly focus on the ordinal patterns within data sequences, often overlooking the embedded amplitude variations that carry critical information. This study introduces a novel complexity measure, termed amplitude-sensitive permutation entropy (ASPE). ASPE integrates amplitude information by weighting ordinal patterns based on their coefficient of variation, a measure that captures data variability relative to the mean. ASPE's performance was first evaluated through simulations, where it outperformed five existing PE methods in detecting amplitude changes and accurately allocating weights. To confirm its broader applicability, ASPE was subsequently tested on two amplitude-rich physiological time series datasets. The results showed that ASPE not only captures dynamic variations more effectively but also surpasses existing methods in distinguishing between disease states.
\end{abstract}



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\section{Introduction}
\label{intro}
Physiological time series, including electrocardiogram (ECG), electroencephalogram (EEG), blood pressure, respiration, etc., provide dynamic recordings of human physiological activities~\cite{shiAmplitudeModulationMultiscale2023,wang2022multiscaleincremententropy}. Their non-invasive collection methods ensure safety and comfort, while their cost-effectiveness and high spatiotemporal resolution provide rich physiological insights. These attributes make physiological time series indispensable for medical diagnosis, disease monitoring, and scientific research~\cite{wangMultiscaleIncrementEntropy2022}. As data collection technologies advance, the accumulation and dissemination of physiological data continue to grow, fostering global medical research and interdisciplinary collaboration.

The analysis of physiological time series involves various methods, including time-domain analysis, frequency-domain analysis, and nonlinear dynamical methods. Time-domain analysis describes signals by statistical features, while frequency-domain analysis transforms signals into the frequency domain to reveal their distribution characteristics at different frequencies~\cite{deaguiarneto2019}. Due to the inherent complexity and nonlinear dynamical properties of physiological systems, the use of nonlinear methods to analyze physiological time series often yields richer and deeper insights~\cite{shen2024efficientprematureventricular}, aiding in elucidating the intrinsic regularities and mechanisms of physiological systems. In nonlinear analysis, entropy is a widely used tool to characterize the dynamic changes of signals~\cite{spichak2022exploitingimpactordering}. Over the past few decades, various entropy-based methods have been widely used in the analysis of physiological time series, such as sample entropy (SE)~\cite{richmanPhysiologicalTimeseriesAnalysis2000a,cabanas2024}, approximate entropy (AE)~\cite{pincusApproximateEntropyMeasure1991a,sklenarova2023}, and dispersion entropy (DE)~\cite{jiangDispersionComplexityEntropy2024,kafantaris2023}. While these techniques offer valuable quantitative insights into signal randomness and regularity, their application in physiological time series analysis reveals certain limitations. Firstly, SE's sensitivity to data length and noise necessitates sufficiently long time series samples and low noise levels for reliable results~\cite{wang2021}. Additionally, parameter selection plays a critical role; for instance, AE depends on a pre-set error tolerance~\cite{guoIntelligentFaultDetection2022}, while DE's accuracy is influenced by assumptions regarding data distribution~\cite{azami2018}.

The permutation entropy (PE)~\cite{bandt2002} method, introduced by Bandt and Pompe in 2002, constitutes an entropy analysis technique grounded in symbolic processes. By quantifying the frequencies of ordinal patterns derived from state space vectors, PE offers a means of measuring the complexity and dynamical behavior of time series, reflecting the inherent unpredictability and randomness of the data~\cite{yangHierarchicalMultiscalePermutation2021,zhengGeneralizedCompositeMultiscale2018}. This capability has led to its widespread application in fields such as heart rate variability analysis~\cite{carricartenaranjo2017}, EEG signal processing~\cite{zanin2023}, and exercise physiology~\cite{gonzalo2019permutationentropyirreversibility}. However, in the calculation of PE, only the ordinal relationships between data points are considered~\cite{zheng2019compositemultiscaleweighted,jomaa2019multivariateimprovedweighted}, resulting in the disparate treatment of data with varying structures and distributions, which are assigned to the same pattern. To illustrate, despite significant differences in the amplitudes of data points $\left\{ 1,2,3 \right\}$, $\left\{ 1,1.1,3 \right\}$, $\left\{ 1,2.9,3 \right\}$ and $\left\{ 10,20,30 \right\}$, they are all classified into the same ascending pattern. These amplitude variations may contain crucial information within the sequence~\cite{chen2021weightedlinkentropy,zhouUsingMissingDispersion2022}, yet PE disregards this information. In the context of contemporary research, two distinct methodological approaches have emerged as viable solutions to address the challenges. These are the discretization-based and weighting-based approaches. Discretization-based solutions introduce a pre-symbolization discretization process related to amplitude, incorporating amplitude information into the symbolization process. For example, in 2019, Chen et al.~\cite{chen2019} proposed an improved permutation entropy (IPE) method to quantify the complexity of time series under noisy conditions. They introduced a discretization factor and discretized the data based on amplitude information before symbolization, addressing the neglect of amplitude information problem. In 2022, Mansourian et al.~\cite{mansourian2023} extended IPE with adaptive improved permutation entropy (AIPE), which sets adaptive discretization thresholds to avoid identical discretization results from a universal discretization factor. Weighting-based approaches introduce amplitude information differently, maintaining the symbolization process unchanged but assigning amplitude-related weights to vectors during the probability distribution calculation. A representative example is the weighted permutation entropy (WPE) proposed in 2013~\cite{fadlallah2013}, which uses the variance of vectors as weights during the probability calculation of ordinal patterns, thus extracting complexity information from data with mutations. In 2021, Ren et al.~\cite{ren2021} introduced rescaled range permutation entropy (RSPE), specifying the ratio of range to standard deviation as weights for each extracted vector, serving as a new metric to measure the complexity of chaotic time series with extreme fluctuations.

Although these improvement schemes have significantly advanced the understanding of amplitude change and have been applied in various studies~\cite{stosic2024,liNovelSchemeBased2024}, they still fall short in fully resolving existing issues. In the case of IPE, the use of discretization can escalate the number of ordinal patterns from a factorial level to an exponential level, significantly increasing computational overhead. Furthermore, IPE's discretization scale is influenced by data extremes, which can lead to a clustering of results at the boundaries when there is a substantial difference between these extremes, potentially producing misleading outcomes. AIPE addresses this issue by using adaptive thresholds to determine the subsequent discrete values, thereby reducing clustering. However, this adaptive approach further increases the number of ordinal patterns, making the method less efficient. In weighting-based schemes, accurately capturing amplitude dynamics requires more than just detecting these changes; it also necessitates consideration of the magnitude of fluctuations and the inherent complexity of the data. Insufficient or improper weight allocation can lead to substantial biases, potentially distorting the accuracy of the entropy measurement and undermining the reliability of the analysis. In the case of WPE's variance-based weighting method, an increase in data magnitude disproportionately inflates the variance, causing larger values to overshadow the characteristics of smaller magnitude data. This bias toward larger magnitudes can obscure the overall data's features. RSPE mitigates this issue by employing the ratio of data range to standard deviation as weights, a method that accounts for both data dispersion and range, thereby reducing the dominance of large magnitudes and providing a more balanced reflection of data dispersion. However, since the data range only offers insight into the extent of the distribution and does not capture its specific shape, relying solely on range for weight assignment may result in suboptimal outcomes.

After thoroughly evaluating the strengths and limitations of existing methods, this paper introduces a novel amplitude-sensitive permutation entropy (ASPE) method based on the coefficient of variation. ASPE uses the ratio of standard deviation to mean as the weighting factor for vectors, effectively capturing both the average level and dispersion of the data. This approach not only enhances the sensitivity to amplitude changes but also preserves the overall complexity assessment of the data. To validate the effectiveness of ASPE, we conducted experiments on both simulated and real physiological time series datasets. Simulated data scenarios were designed to represent various amplitude changes and compared against five existing PE methods. The results demonstrated that ASPE offers significant advantages in the perception of amplitude change. For real-world validation, we applied ASPE to two EEG datasets with epileptic seizures, one ECG dataset with arrhythmic patterns, and another ECG dataset depicting sleep apnea. These datasets, rich in amplitude variations, encapsulate the complex dynamic behaviors of physiological systems. The experimental results show that ASPE excels in capturing dynamic information linked to disease states and severity, outperforming other methods. Specifically, ASPE accurately identifies the onset and termination of epileptic seizures, and distinguishes patients with arrhythmia from healthy individuals.

% 本文以下各节的结构如下： 第 2 章详细介绍了所提出的置换熵方案。第3章对实验设置进行了说明。随后，第 4 章在合成数据上对ASPE的性能进行了验证与分析。第 5 章介绍了ASPE在真实数据中的表现。最后，第6 章对整篇论文进行了总结。
The following sections of this paper are structured as follows: Chapter~\ref{sec:method} provides a detailed introduction to the proposed permutation entropy scheme. Chapter~\ref{sec:setup} provides a detailed explanation of the experimental setup. Subsequently, Chapter~\ref{sec:syn} presents a validation and analysis of the performance of ASPE on synthetic data. Chapter~\ref{sec:real} presents a detailed analysis of the performance of ASPE in real data sets. Finally, Chapter~\ref{sec:con} offers a summary of the entire paper.


\section{Method}
\label{sec:method}

Based on the analysis of existing approaches , this study propose an innovative weighted improvement of the permutation entropy method, termed amplitude-sensitive permutation entropy (ASPE). ASPE, grounded in the principles of information theory, quantifies the system's randomness or uncertainty by evaluating the frequency of ordinal patterns in time series data. The calculation process of ASPE involves four key stages: phase space reconstruction, ordinal pattern symbolization, probability calculation, and entropy calculation, as illustrated in Fig.~\ref{fig:flow}~(a). The detailed steps of this process are outlined below.

\begin{figure*}[!htbp]
  \centering
  \includegraphics[width=1\linewidth]{flow.pdf}
  \caption{\label{fig:flow}The calculation process of ASPE and examples of ordinal patterns. (a) The calculation process of ASPE, which can be divided into four steps: phase space reconstruction, symbolization, probability calculation, and entropy calculation. (b) Examples of two numerical values with two different ordinal patterns. (c) Examples of six ordinal patterns for three numerical values.}
  \end{figure*}


In line with Takens' Delay Embedding Theorem~\cite{peng2024,wangMultiscaleDiversityEntropy2020}, a phase space identical to the original dynamical system can be reconstructed from a one-dimensional time series. The reconstruction process involves two key parameters: the embedding dimension $d$ and the time delay $\tau$. Give a time series $X=\{{x_{t}\}}$ with length $T$, its phase space representation can be formulated as follows:

\begin{align}
\label{eq:eq1}
{Y}^{d,\tau}=\left[ \begin{matrix}
   y_{1}^{d,\tau }  \\
   y_{2}^{d,\tau }  \\
   \vdots   \\
   y_{i-1}^{d,\tau }  \\
   y_{i}^{d,\tau }  \\
\end{matrix} \right]=\left[ \begin{matrix}
   {{x}_{1}} & {{x}_{1+\tau }} & \cdots  & {{x}_{1+\left( d-2 \right)\tau }} & {{x}_{1+\left( d-1 \right)\tau }}  \\
   {{x}_{2}} & {{x}_{2+\tau }} & \cdots  & {{x}_{2+\left( d-2 \right)\tau }} & {{x}_{2+\left( d-1 \right)\tau }}  \\
   \vdots  & \vdots  & \vdots  & \vdots  & \vdots   \\
   {{x}_{i-1}} & {{x}_{i-1+\tau }} & \cdots  & {{x}_{i-1+\left( d-2 \right)\tau }} & {{x}_{i-1+\left( d-1 \right)\tau }}  \\
   {{x}_{i}} & {{x}_{i+\tau }} & \cdots  & {{x}_{i+\left( d-2 \right)\tau }} & {{x}_{i+\left( d-1 \right)\tau }}  \\
\end{matrix} \right]
\end{align}
where $1\le i\le T-\left( d-1 \right)\tau =N$. Each row in ${{Y}^{d,\tau }}$ represents a phase space vector, composed of $d$ data points with a sampling interval of $\tau $.

For a vector of $d$ elements, there are $d!$ possible ordinal relations, denoted by $\left\{\pi_{j}^{d},1\le j\le d! \right\}$, each representing a unique ordinal pattern (OP). Fig.~\ref{fig:flow}~(b) illustrates two OPs for $d=2$, while Fig.~\ref{fig:flow}~(c) enumerates six OPs for $d=3$. The order of the values in each vector within the phase space can be represented by a unique ordinal pattern, denoted as $type(.)$. This mapping process transforms raw data from phase space into symbol space, effectively symbolizing the data. However, by relying solely on the ordinal relationships, this approach overlooks the equally important magnitude information between data points. To address this, our study incorporates the coefficient of variation (CV) to capture the magnitude changes within vectors, thus integrating magnitude information into the complexity measurement process. Specifically, the coefficient of variation is calculated as follows:

\begin{align}
  \label{eq:eq10}
  {{\omega }_{i}}=\frac{std\left( y_{i}^{d,\tau } \right)}{mean\left( y_{i}^{d,\tau } \right)}
\end{align}

Subsequently, the probability $p\left( \pi _{j}^{d} \right)$ of each OP in the sequence of symbols is calculated based on the number of occurrences of each OP and the weight of each phase space vector.

\begin{align}
  \label{eq:eq2}
  p\left( \pi _{j}^{d} \right)=\frac{\sum\limits_{i=1}^{N}{\left[ type\left( y_{i}^{d,\tau } \right)=\pi _{j}^{d} \right]{{\omega }_{i}}}}{\sum\limits_{i=1}^{N}{{{\omega }_{i}}}}
  \end{align}
where the function $\left[ . \right]$ returns 1 if the given statement is true, and 0 otherwise. If all weights of the vector are equal to one, the ASPE reduces to a conventional PE. Once the probability of each OP has been calculated, the ordered symbolic probability distribution, represented by $P^{d}$ can be obtained.
The next step is to calculate the Shannon entropy of the probability distribution, which can be done using the following formula:

  \begin{align}
  \label{eq:eq3}
  S\left( {{P}^{d}} \right)=-\underset{j=1}{\mathop{\overset{d!}{\mathop{\sum }}\,}}\,p\left(\pi _{j}^{d} \right)\log p\left( \pi _{j}^{d} \right)
  \end{align}

The value of $S\left( {{P}^{d,\tau }} \right)$ varies with changes in the embedding dimension $d$, reaching its maximum when ${{P}^{d,\tau }}$ follows a uniform distribution. To facilitate the comparison of entropy values across different parameters, it is customary to normalize the entropy by its maximum possible value for the given embedding dimension $d$:

\begin{align}
\label{eq:eq4}
ASPE=\frac{S\left( {{P}^{d}} \right)}{S\left( P_{e}^{d} \right)}
\end{align}
  where $P_{e}^{d}$ denotes a uniformly distributed probability vector consisting of $d!$ equal probability values of $1/d!$ each. The normalized permutation entropy value thus lies between 0 and 1. The detailed computational method of ASPE is presented in Algorithm~\ref{alg1}.

\begin{algorithm}[!htbp]
        \SetAlgoLined
          \caption{ASPE}
          \label{alg1}

          \KwIn{Time series $X=\{x_t,1\le t\le T\}$, embedding dimension $d$, time delay $\tau$.}
          \KwOut{ASPE.}
            \tcp{Normalize data to a standard range.}
          $ \hat{X} = [{X - \text{min}(X)}]/[{\text{max}(X) - \text{min}(X)}]$  \;
            \tcp{Reconstruct the phase space.}
            $Y^{d,\tau}=\{ y_i^{d, \tau},1\le i\le N  \}$, $N=T-(d-1)\tau$\;
            \For{$i=1$ to $N$}{
                $y^{d,\tau}_{i}=\{\hat{x}_{i},\hat{x}_{i+\tau},...,\hat{x}_{i+(d-2)\tau},\hat{x}_{i+(d-1)\tau}\}$\;
          }
            \tcp{Calculate weights.}
            $W^d=\{weight_j^d, 1\le j \le d!\}$\;
            \For{$i=1$ to $N$}{
                $\omega_{i}={std(y^{d,\tau}_{i})}/{mean(y^{d,\tau}_{i})}$\;
              \For{$j=1$ to $d!$}{
                    \If{$type(y^{d,\tau}_{i})=\pi_{j}^{d}$}{
                        $weight_j^d=weight_j^d+\omega_{i}$\;
                      }
                }
          }
            \tcp{Calculate probabilities.}
            $P^{d,\tau}=\{ p(\pi_{j}^{d}), 1 \le j \le d! \}$\;
            \For{$j=1$ to $d!$}{
                $p(\pi_{j}^{d})={weight_j^{d}}/{sum(W^{d})}$\;
            }
            \tcp{Calculate Shannon entropy.}
            $S(P^{d,\tau}) = 0 ,S(P_{e}^{d}) = 0 $\;
            \For{$j=1$ to $d!$}{
                $S\left ( P^{d,\tau}  \right )=S\left ( P^{d,\tau}  \right )-p (\pi^{d}_{j}) log p(\pi^{d}_{j})$\;
                $S\left ( P^{d}_{e}  \right )=S\left ( P^{d}_{e}  \right )-\frac{1}{d!}  log \frac{1}{d!}$\;
            }
            \tcp{Compute ASPE.}
            $ASPE(d,\tau)={S\left ( P^{d,\tau}  \right )}/{S\left ( P^{d}_{e}  \right )} $\;
\end{algorithm}


% CV 作为一个无量纲指标，通过计算标准差与平均值的比值来表征原始数据的离散程度。这一指标不仅受变量值变化的影响，也受其平均值大小的影响。与离散化方案相比，ASPE 不需要额外的参数，不会增加序数模式的数量，并保持相同的计算复杂度。与现有的基于加权的方法相比，ASPE 在感知振幅变化和权重分配方面更具合理性。具体来说，我们使用一个包含振幅变化的模拟数据来说明基于加权法的权重分配，如表所示。
The CV, as a dimensionless indicator, characterizes the dispersion of original data by calculating the ratio of the standard deviation to the mean. This metric is influenced not only by the variability of the variable values but also by the magnitude of their mean values. Compared to discretization schemes, ASPE requires no additional parameters, does not increase the number of ordinal patterns, and maintains the same computational complexity. Compared to existing weighting-based approaches, ASPE demonstrates greater rationality in the perception of amplitude change and weight allocation. Specifically, the weight allocation of weighting-based approaches is illustrated using one simulated datas containing amplitude variations, as shown in Table~\ref{tab:table1}.

\begin{table}[!htbp]
\centering
\caption{Weight allocation results on simulated amplitude data.}
\label{tab:table1}
\begin{ruledtabular}
\begin{tabular}{ccc}
\multirow{2}{*}{Entropy} & \multicolumn{2}{c}{$X=\{1,2,3,24\}$}  \\
\cmidrule(lr){2-3}
& $\omega_1$\footnote{$\omega_i$ represents the weight of $y_{i}^{3,1}$.} & $\omega_2$ \\
\hline
WPE & 1~(0.64\%)\footnote{The number in brackets represents the weight as a percentage of all weights.} & 154~(99.35\%) \\
RSPE & 2~(53.05\%) & 1.77~(46.94\%) \\
ASPE & 0.5~(28.08\%) & 1.28~(71.91\%)
\end{tabular}
\end{ruledtabular}
\end{table}

%% 从权重分配的模拟结果上看，WPE 对发生振幅变化的数据w2赋予了过高的权重，导致数据其他部分的权重不成比例地减弱。与之相比，虽然 RSPE 为w1分配了较小的权重，但这些权重与w1的权重过于接近，不可避免的使得 RSPE 有退化为 PE 的趋势。ASPE 的权重介于 WPE 和 RSPE 之间，既能有效强调振幅变化的重要性，又能保持其他部分权重的合理代表性。

From the simulation results of weight assignment, WPE assigns excessively high weights to data undergoing amplitude changes, leading to a disproportionate weakening of weights in other portions of the data. Conversely, while RSPE assigns smaller weights to data in the amplitude-changing regions, these weights are too close to those assigned to the main portion, causing RSPE to tend towards PE. ASPE's weighting falls between those of WPE and RSPE, effectively emphasizing the importance of amplitude changes while maintaining a reasonable representation of weights in other parts.

%% ======================

\section{Experimental setup}
\label{sec:setup}

In terms of experimental parameter settings, a consistent setup was maintained, following similar studies~\cite{liNovelSchemeBased2024,guoIntelligentFaultDetection2022}. Specifically, the embedding dimension was set to 5, the time delay was set to 1, and the data length followed $T\gg \left( d+1 \right)!$, unless otherwise stated. Additionally, the scale factor parameters in IPE and AIPE followed the settings recommended in their respective research papers. All the code used in this study was executed on a computer with the following configuration: Intel Core i5-11400 CPU (2.59 GHz), 16.0 GB RAM, and a Windows 10 operating system. The programming environment used was MATLAB R2022b.


To ensure the rigor and reliability of the experimental results, several measures were taken to reduce experimental error and randomness:

\begin{enumerate}

\item For simulated data, $100$ data sets were generated, and their averages were used for quantitative analysis to reduce the variability of individual experiments and improve the stability of the results.
\item To further validate the findings, all significance level test results were corrected using Bonferroni correction. ``ns'' indicates no significant difference, ``$*$'' indicates $P\le0.05$, ``$**$'' indicates $P\le0.01$, and ``$***$'' indicates $P\le0.001$.
\item For classification tasks, six representative machine learning classifiers (Random Forest, Decision Tree, Support Vector Machine, Naive Bayes, K-Nearest Neighbors, and Logistic Regression) were used for 10 iterations of 10-fold cross-validation, with results averaged over each iteration to increase reliability.
\end{enumerate}




%% ======================
\section{Synthetic data experiments}
\label{sec:syn}

% 本节中，将基于经典动力学Logistic混沌系统来对ASPE的幅度感知能力进行验证。并与典型的5个排列熵方案进行对比，包括传统PE，离散化方案IPE，AIPE和权重类方案WPE和RSPE。Logistic动力学的系统描述如下：
This section will validate the amplitude-awareness capability of ASPE based on the classical dynamics of the logistic chaotic system. Furthermore, it will compare ASPE with five typical alignment entropy schemes, including traditional PE~\cite{bandt2002}, discretization schemes IPE~\cite{chen2019}, AIPE~\cite{mansourian2023} and weighting class schemes WPE~\cite{fadlallah2013} and RSPE~\cite{ren2021}. The system description of the logistic dynamics is as follows:

The Logistic chaos system is a classical dynamical model frequently used to simulate the growth of biological populations or the oscillatory behavior of circuits~\cite{hernandezFourierPhaseIndex2024}.
\begin{align}
\label{eq:eq11}
{{x}_{n+1}}=r{{x}_{n}}\left( 1-{{x}_{n}} \right)
\end{align}
where $r\in \left[ 0,4 \right]$ is the control parameter that determines the behavioral characteristics of the system. When $r$ exceeds 3.5699, the system enters a chaotic regime.




\subsubsection{Sensitivity to changes in magnitude}

% 本节对ASPE对幅度变化的敏感性进行检验。具体而言，我们在参数alpha设定为4的条件下，利用Logistic混沌系统生成了一个长度为300的数据点序列，该序列呈现出典型的混沌行为。为了模拟幅度变化，我们在序列的第1500个数据点处引入了一个大小为20的脉冲信号,如图1（a）所示。为了量化分析该脉冲信号对系统排列熵的影响，采用大小为1000的滑动窗口技术计算整个数据长度下各个排列熵的值，结果如图1（b）所示。由于滑动窗口的计算方式，模拟信号500~1500个数据点的熵值计算中会包含脉冲信号,记该区域即为impulse-affected area (IAA)。
This section examines the sensitivity of ASPE to amplitude variations. Specifically, using the Logistic chaotic system with the parameter alpha set to $4$, we generated a sequence of $10,000$ data points exhibiting typical chaotic behavior. To simulate an amplitude change, a pulse signal with a magnitude of $20$ was introduced at the $7,500th$ data point in the sequence, as shown in Fig.~\ref{fig:single}~(a). To quantify the impact of this pulse signal on the system’s permutation entropy, we employed a sliding window technique with a window size of $5,000$ to calculate the permutation entropy values across the entire sequence. The results are displayed in Fig.~\ref{fig:single}~(b). Due to the sliding window calculation, the pulse signal is included in the entropy calculation for the data points between $2,500$ and $7,500$, which we define as the impulse-affected area (IAA), marked as a gray region.

\begin{figure}[!htbp]
\centering
\includegraphics[width=1\linewidth]{single.pdf}
\caption{Spike detection application results of six permutation entropy methods. (a) Chaotic signal generated by the Logistic chaotic system. A pulse signal with a value of $20$ was introduced at the $7,500th$ data point to simulate amplitude transitions. (b) Variation of permutation entropy values calculated using a sliding window approach on the simulated signal. The sliding window has a length of $5,000$ and a step size of $10$.}
\label{fig:single}
\end{figure}

For the amplitude change caused by the pulse, PE shows no meaningful change throughout the region, a behavior also observed in RSPE. However, other permutation entropy metrics detect the amplitude change upon entering the IAA, show a decrease in entropy values, and recover after leaving. Specifically, ASPE shows the smallest amplitude change, decreasing by about $3\%$, WPE decreases by about $25\%$, AIPE decreases by about $55\%$, and IPE decreases to nearly 0. The degree of decrease exhibited by different entropy values essentially reflects their sensitivity to amplitude changes. From the calculation process of Shannon entropy, we can conclude that as the entropy value decreases, the probability distribution becomes more concentrated. Especially for IPE and AIPE, their probability distributions may converge on individual patterns, directly resulting in smaller entropy values or even approaching zero.

% 进一步的，我们从熵值的计算过程探讨导致这一结果的原因。为方便展示，选择嵌入维度为3，延迟为1（该参数下的熵值有着与前文相似的变化趋势），以合成数据的IAA区域数据为例，计算了各个有序模式的概率分布，如图7（a）所示。其中红色柱状代表脉冲所属的递增模式的概率值。需要注意的是，IPE和AIPE的离散化方案有着超出3！个的概率分布，此处只展示前6个。
Further analysis of the entropy calculation process reveals the underlying reasons for this result. For clarity, we selected an embedding dimension of 3 and a delay of 1 (parameters that produce a similar trend in entropy values as discussed earlier) and used the IAA region data from the synthetic dataset as an example. The probability distribution of each ordinal pattern $\pi_{1}^{3}$ was computed, as shown in Fig.~\ref{fig:single_detail}~(a). The red bars represent the probability values of the increasing pattern associated with the pulse. It is important to note that the discretization schemes of IPE and AIPE result in more than six possible probability distributions, but only the first six are presented here for simplicity.

\begin{figure}[!htbp]
\centering
\includegraphics[width=1\linewidth]{single_detail.pdf}
\caption{Probability distribution and weight allocation results in the pulse region.(a) Probability distribution results of six types of permutation entropy in the pulse region. (b) Weighting results of $\pi_{1}^{3}$ for the weighting permutation entropy method.}
% 脉冲区域上的概率分布和权重分配结果。（a）六种排列熵在脉冲区域的概率分布结果。（b）权重类排列熵方法的 $\pi_{1}^{3}$ 模式的权重分配结果。
\label{fig:single_detail}
\end{figure}

% 根据香农熵的定义，概率分布的集中程度与熵值之间存在直接关联：概率分布越集中，熵值越小；概率分布越均匀，熵值越大。这一原理在各类熵值的概率分布情况与IAA区域的熵值对比中得到了验证。IPE的离散化方案是根据整体数据的范围来划分的，导致大部分数据被归类到同一个模式$\pi_{1}^{3}$中.这种归类方式使得$\pi_{1}^{3}$出现的概率非常接近100%，从而使得概率分布中的不确定程度显著降低，得到0附件的熵值。
According to Shannon entropy's definition, there is a direct relationship between the concentration of probability distribution and the entropy value: the more concentrated the probability distribution, the lower the entropy; the more uniform the probability distribution, the higher the entropy. This principle is verified by comparing the probability distributions of various entropy measures with the entropy values in the IAA region. The discretization scheme of IPE is based on the overall range of the data, which causes most of the data to be classified into a single pattern, $\pi_{1}^{3}$. This classification method results in the probability of $\pi_{1}^{3}$ approaching $100\%$, significantly reducing the uncertainty in the probability distribution and yielding an entropy value close to zero.
% AIPE的动态离散化方案削弱了这种现象，但其主要的概率仍然集中在$\pi_{1}^{3}$，且部分模式占比为0，熵值较小。权重类方案中，WPE中的pi1概率超过了60%，而其他模式的概率则显著减小了;而RSPE在各个模式上都得到了非常接近于PE的概率分布。相比之下，而ASPE则没有给pi分配过多的权重，pi4和pi5的权重相比PE有一定的上升，概率分布变得更加集中。脉冲的影响使得ASPE的熵值较PE有一定的下降，但并未导致其他模式的重要性被忽略。
The dynamic discretization scheme of AIPE mitigates this effect, but the probability remains primarily concentrated in $\pi_{1}^{3}$, with some patterns having a probability of zero, resulting in a relatively low entropy. In the weighted entropy methods, WPE assigns more than $60\%$ probability to the $\pi_{1}$ pattern, while the probabilities of other patterns are significantly reduced. In contrast, RSPE achieves a probability distribution that is very close to that of PE across all patterns. ASPE, on the other hand, does not assign excessive weight to $\pi_{1}$; instead, the weights of $\pi_{4}$ and $\pi_{5}$ increase slightly compared to PE, making the probability distribution more concentrated. Although the pulse's influence causes ASPE's entropy to decrease slightly compared to PE, it does not lead to the neglect of other patterns.

% 此外，我们还注意到，各个熵值都出现了递减有序模式$\pi_{6}^{3}$的缺失，这种确实的缺失模式已被证实是动力学系统确定性的一种表征，但此处不做讨论。
Additionally, we observe that all entropy measures exhibit the absence of the decreasing ordinal pattern $\pi_{6}^{3}$. This missing pattern has been confirmed as a representation of the determinism in dynamical systems~\cite{du2020detectinggasliquid}, though it will not be discussed further here.

% 从概率分布的结果中分析得出，WPE给pi1模式分配了过高的权重，而RSPE的分配方式则退化成了接近PE的程度。为了验证这一推论，我们记录了IAA中每个pi模式的权重分配情况，如图7（b）所示。其中w_x是p1模式占所有pi1模式权重和的比例，w_y是pi模式占所有模式的权重和的比例,并使用☆标记了脉冲信号所在模式的权重。

From the probability distribution results, we conclude that WPE assigns excessive weight to the $\pi_{1}$ pattern, while RSPE's distribution has regressed to a level close to that of PE. To validate this hypothesis, we recorded the weight allocation of each $\pi$ pattern in the IAA, as shown in Fig.~\ref{fig:single_detail}~(b). Here, $w_x$ represents the proportion of the $\pi_{1}$ pattern's weight to the total weight of all $\pi_{1}$ patterns, and $w_y$ represents the proportion of the $\pi$ pattern's weight to the total weight of all patterns. The pulse signal’s pattern is marked with a $\bigstar$ symbol.

% 从权重分配的结果中，我们找到了支撑上述结论的重要证据。PE的权重分配结果显示，pi1模式的数量共有500个，均匀分配的权重在0.35%左右，占整体有序模式的0.15%左右。而在WPE中，脉冲部分的方差权重占据了所有pi的80%，在所有有序模式中也占据了50%。脉冲部分的权重对整个数据的熵值起到了决定性的作用，而其他有序模式的存在感被显著削弱。相比之下，RSPE中的wx和wy在均匀分布的范围内呈小幅度变化，脉冲点呈现出下降的趋势。整体上，RSPE的权重分配与均匀分配的差距不大，这是RSPE退化为PE的根本原因。本文提出的ASPE则介于WPE和RSPE之间，在wx中,脉冲点的权重上升呢0.1%，在wy中上升了0.01%。这种权重分配方式既反映了脉冲带来的幅度变化，又没有对其他模式造成过度的影响。
The weight allocation results provide key evidence supporting the above conclusions. The weight allocation of PE shows that the $\pi_{1}$ pattern appears 500 times, with evenly distributed weights around $0.35\%$, accounting for approximately $0.15\%$ of all ordinal patterns. In WPE, the variance weight of the pulse section occupies $80\%$ of all $\pi$ weights and $50\%$ of all ordinal patterns. The weight of the pulse section plays a decisive role in the entropy of the entire dataset, significantly diminishing the presence of other ordinal patterns. In contrast, the $w_x$ and $w_y$ values in RSPE show slight variations within the uniform distribution range, and the weight of the pulse point shows a downward trend. Overall, RSPE's weight allocation does not differ significantly from the uniform distribution, which is the primary reason RSPE regresses to PE. ASPE, positioned between WPE and RSPE, shows a $0.1\%$ increase in the pulse point's weight in $w_x$ and a $0.01\%$ increase in $w_y$. This weight allocation reflects the amplitude variation caused by the pulse without disproportionately affecting other patterns.

\subsubsection{Multi-Amplitude Sensitive Results}

To simulate the diversity of amplitude changes in real data, we constructed simulated data containing multiple amplitude transition points. This aimed to comprehensively evaluate the performance of the ASPE method in handling complex amplitude variations. The simulated signals are illustrated in Fig.~\ref{fig:mult}~(a). The entropy variation results of the six permutation entropy methods are presented in Fig.~\ref{fig:mult}~(b).

\begin{figure}[!htbp]
\centering
\includegraphics[width=1\linewidth]{mult.pdf}
\caption{ Experimental results of multi-amplitude variation Sensitive. (a) Simulated signal with multiple amplitude variations. Generated by the Logistic chaotic system, pulses of different amplitudes are uniformly added to the signal. The shaded areas represent the IAA, numbered from 1 to 9. (b) Entropy variation results of six permutation entropy methods applied to the simulated signal. (c) Significance analysis results of the six permutation entropy methods between adjacent IAAs. The number of IAAs is expanded to 20, with pulse amplitudes ranging from 10 to 200, in steps of 10. The average entropy value within each IAA is calculated across 100 simulated signals, and significance tests are conducted between adjacent IAAs.}
\label{fig:mult}
\end{figure}

PE shows a stable trend over the entire range of data, with no significant fluctuations in entropy values as a function of amplitude. Similarly, RSPE reaches conclusions similar to PE, with even smoother trends. We speculate that this may be due to the mutual neutralization of the weight distribution in the amplitude change region and the original complexity change. IPE and AIPE rapidly decrease to levels near zero at the first amplitude change and remain relatively stable during subsequent amplitude changes, failing to show significant differences. This extreme sensitivity to amplitude changes makes it difficult for them to effectively discriminate between different degrees of amplitude change. WPE shows significant entropy fluctuations in the first three amplitude change intervals, reaching a minimum value of about 0.38 in the fourth interval, but loses significant discriminative power thereafter. This suggests that WPE has some resolution within a certain amplitude range, but its effectiveness is significantly limited when faced with larger amplitude changes. In contrast, our proposed ASPE shows a slight decrease in entropy values in the first amplitude change interval and a continuous downward trend with increasing amplitudes. Importantly, there are significant differences in entropy values between adjacent amplitudes, allowing ASPE to more accurately identify and distinguish between different amplitude changes.

Furthermore, the range of amplitude changes was expanded to 20 steps, and the significance tests on the entropy values of adjacent pairs of results are shown in Fig.~\ref{fig:mult}~(c). PE, IPE, AIPE, and RSPE showed no significant differences in any of the comparison groups. WPE exhibited some discriminative ability in the first three comparison groups with significant differences, but its effectiveness was significantly limited in subsequent larger amplitude changes. In contrast, our proposed ASPE showed significant differences in entropy values in as many as 13 comparison groups, further highlighting the outstanding performance of ASPE in discriminating between different amplitude changes.


\subsection{Consistency and robustness}

% 在排列熵最初的研究当中，对于振幅变化具有不敏感性是排列熵的一种理想特性，尤其是在存在观测噪声的情况下，这种不敏感性赋予了排列熵对噪声的鲁棒性。本文提出的具有幅度感知的排列熵并不是对这一理想特性的背道而驰,而是一种探索复杂系统内在确定性的一种功能加强。ASPE能够在不削弱排列熵的这种特性的情况下从幅度信息中挖掘更多管理复杂系统动力学行为的信息。为了验证ASPE的有效性和其对排列熵特性的保留情况，本文使用模式数据从一致性和鲁棒性两个角度来验证ASPE对这一特性的保留。详见附录A。
In the initial research on permutation entropy (PE), its insensitivity to amplitude variations was considered an ideal characteristic, particularly in the presence of observational noise, as this insensitivity granted PE robustness against noise. The amplitude-sensitive permutation entropy (ASPE) proposed in this paper does not deviate from this ideal characteristic. Rather, it enhances the functionality of PE by enabling the extraction of additional information about the dynamics of complex systems from amplitude data, without compromising PE's inherent noise robustness. To validate the effectiveness of ASPE and its preservation of PE's key characteristics, we employed simulated data to assess its consistency and robustness from both perspectives.

% 一致性指的是的是熵的数值变化与复杂系统的动力学行为是否保持一致。下面基于Logistic混沌系统对ASPE的一致性进行考察。设置a的参数范围为3.5~4,其分叉图如图7A所示。然后，采用了参数m=5，d=1，并选取了5000个数据点作为样本，以0.001为步长，分别计算PE和ASPE在不同混沌参数a下的熵值变化趋势，并与Logistic的李雅普诺夫指数进行对比，结果如图7B和C所示。

\subsubsection{Consistency}
\label{sec:sec_consi}

Consistency refers to whether the changes in entropy values align with the dynamical behavior of complex systems. To examine the consistency of ASPE, we conducted an analysis based on the Logistic chaotic system. The parameter $a$ was set within the range of 3.5 to 4, with its bifurcation diagram shown in Fig~\ref{fig:consi}~(a). Using parameters $m=5$, $d=1$, and a sample size of 5,000 data points, we calculated the trends of both PE and ASPE entropy values at different chaotic parameter values, with a step size of 0.001. The results were then compared with the Lyapunov exponent of the Logistic system, as illustrated in Fig~\ref{fig:consi}~(b) and (c). In the Lyapunov exponential diagram, the system is deemed to have entered a chaotic state when the Lyapunov exponent is greater than zero, and conversely, a cyclic state when it is less than zero. Furthermore, for purposes of comparison, the three parameter ranges exhibiting notable kinetic behavioural alterations are indicated by red boxes.

\begin{figure}[!htbp]
\centering
\includegraphics[width=1\linewidth]{consi.pdf}
\caption{Consistency test results for Logistic chaotic systems with different parameters. (a) Bifurcation diagram. (b) Lyapunov exponent. (c) PE and ASPE.}
\label{fig:consi}
\end{figure}

The results of the entropy change analysis of ASPE demonstrate a striking similarity in its trend when compared to PE. In particular, during the cycle phase of the logistic system, the entropy change trends of ASPE and PE exhibit near-complete overlap. In contrast, during the chaos phase, the entropy value of ASPE displays a slight decline in comparison to PE, yet both continue to exhibit analogous fluctuation patterns. When the chaotic dynamics underwent significant alteration, ASPE exhibited corresponding behavioural characteristics that reflected the changes observed in the bifurcation diagram and the Lyapunov exponent plot. Overall, ASPE's performance in the logistic mapping system did not deteriorate due to sensitivity to magnitude and underwent degradation, while still maintaining a similar ability to PE.

\subsubsection{Robustness}
\label{sec:sec_rob}


% 排列熵的鲁棒性体现在其对噪声的抵抗能力和对数据长度的低敏感性上.这一特性主要归因于排列熵算法的核心机制：它专注于数据序列中元素之间的序数关系（即相对顺序），而非具体的数值大小。为了深入验证ASPE的上述特性,本节基于Logistic混沌系统生成的信号进行考察,alpha取值为4(系统进入混沌状态).在验证过程中，我们遵循上文选择了嵌入维度为5和延迟时间为1的参数设置，并且为了量化ASPE在抵抗噪声能力和数据长度要求方面的表现，我们引入了变化率指标。变化率指标的计算方式如下：

The robustness of permutation entropy is reflected in its resistance to noise and low sensitivity to data length. This property is mainly attributed to the core mechanism of the alignment entropy algorithm: it focuses on the ordinal relationship between elements in the data sequence rather than the specific numerical size. In order to verify in depth the above property of ASPE, this section examines it on the basis of the signals generated by a logistic chaotic system where alpha takes the value $4$ (the system enters a chaotic state). In the validation process, we followed the above and chose the parameter settings of embedding dimension of $5$ and delay time of $1$. In addition, to quantify the performance of ASPE in terms of resistance to noise and data length requirement, we introduced the rate of change (RC)~\cite{yangHierarchicalSymbolTransition2022}. The RC is calculated as follows:

\begin{align}
\text{RC} & = \frac{|H - H_{\text{0}}|}{H_{\text{0}}} \times 100\%
\end{align}
where $H$ represents the entropy value under the current noise and $H_0$ represents the entropy value in the case of no noise.

% 首先验证ASPE对噪声的抵抗能力。使用混沌系统生成长度为5000的混沌序列，加入不同信噪比（SNR=40:-1:0 dB）的高斯白噪声，计算PE和ASPE的熵值变化率。重复此过程100次，结果如图7A所示。

% 然后验证ASPE对数据长度的要求。使用混沌系统生成长度为20000的混沌序列，从中截取不同长度的型号（Length=20000:-500:500），计算PE和ASPE的熵值变化率。重复此过程100从，结果如图7B所示。

The resistance of ASPE to noise was initially validated. A chaotic sequence of length $5000$ was generated using the chaotic system. Gaussian white noise with different signal-to-noise ratios (SNR) (ranging from $40$ to $0$, with a step size of $1$) was then added, and the entropy change rates of PE and ASPE were calculated. This process was repeated $100$ times, and the results are presented in Fig.~\ref{fig:robust}~(A). Subsequently, the ASPE is validated with respect to the requisite data length. Utilising the chaotic system, a chaotic sequence of length $20,000$ is generated, from which models with varying lengths (from $20,000$ to $5,000$, with decrements of $500$) are extracted to compute the entropy value change rate of PE and ASPE. This process is repeated $100$ times, and the resulting data are presented in Fig.~\ref{fig:robust}~(B).

\begin{figure}[!htbp]
\centering
\includegraphics[width=1\linewidth]{robust.pdf}
\caption{Robustness experiment results of ASPE. (a) Entropy change rate of ASPE under the influence of Gaussian white noise with different signal-to-noise ratios. (b) Entropy change rate of ASPE under the influence of different data lengths.}
\label{fig:robust}
\end{figure}

% 从噪声水平和不同数据长度下的变化率表现看来，ASPE有着非常接近PE的鲁棒性表现。具体来说，在高信噪比条件下，ASPE有着低于PE的变化率，展现出良好的稳定性能。但是随着信噪比的降低，模拟信号中的噪声占比逐渐增大，ASPE与PE之间的变化率差异开始显现。特别是在SNR低于11之后，ASPE的变化率开始超越PE，并在SNR为0时，其变化率相较于PE高出约1%；在数据长度实验中，我们也观察到了类似的现象。在10000及更高的数据长度下，ASPE与PE的变化率几乎一致，表现出高度的稳定性。然而，当数据长度降低至10000至2500的范围内时，PE的变化率开始逐渐超过ASPE。但随着长度的进一步降低，PE的高鲁棒性开始显露，ASPE的变化率增大，在最低长度500的情况下比PE高了0.09%。

In terms of noise level and rate of change performance under different data lengths, it appears that ASPE exhibits a robust performance that is nearly equivalent to that of PE. Specifically, under conditions of high SNR, ASPE exhibits a lower rate of change than PE, indicating good stability. However, as the SNR decreases, the proportion of noise in the analogue signal gradually increases, and the difference in the rate of change between ASPE and PE begins to emerge. In particular, when the SNR is below 11, the rate of change of ASPE exceeds that of PE by approximately $1\%$. This is observed to be the case even when the SNR is as low as 0. A similar phenomenon is observed in the data length experiments. At data lengths of $10,000$ and above, the change rates of ASPE and PE are observed to be almost identical, indicating a high degree of stability. Nevertheless, as the data length decreases to the range of $10,000$ to $2,500$, the rate of change of PE begins to exceed that of ASPE. However, as the length decreases further, the high robustness of PE becomes evident, and the rate of change of ASPE increases, outperforming PE by $0.09\%$ at the shortest length of 500.

% 总体说来，在鲁棒性检验中，PE在高噪声和低数据长度下有着最佳的表现.然而，ASPE同样表现出色，尤其是在极端条件下时，其变化率仅比PE略高.这表明ASPE对数据幅度的引入并未显著降低其鲁棒性.

In general, PE demonstrates the most robust performance in the robustness test under conditions of high noise and low data length. However, ASPE also exhibits a commendable degree of resilience, particularly in extreme circumstances, with a rate of change that is only marginally higher than that of PE. This indicates that the incorporation of data magnitude into ASPE does not markedly diminish its robustness.




%% =============================================

\section{Real-world data experiments}
\label{sec:real}

% 在真实数据实验中，本文选择了一个脑电图癫痫数据集和一个心电图心律失常数据集。这些临床数据集具有广泛的应用价值，可以全面评估我们提出的方案在实际应用中的有效性。在使用前，我们对真实数据集进行了标准预处理，包括滤波操作，以消除噪声干扰。
For experiments with real data, one EEG epilepsy datasets and one ECG arrhythmia dataset were selected. These clinically obtained datasets have broad application value and can thoroughly evaluate the effectiveness of our proposed scheme in practical applications. Standard preprocessing procedures, including filtering operations, were applied to the real datasets to eliminate noise interference before use.

\subsection{Epilepsy EEG dataset}

% 波恩癫痫数据库（Bonn Epilepsy Database~\cite{andrzejak2001}）源自波恩大学，由 5 名健康受试者和 5 名癫痫患者的脑电图数据组成。数据收集过程中使用了 128 通道脑电图系统，按照标准的 10-20 电极放置方式，以 173.61 Hz 的采样率进行记录。值得注意的是，该数据集不包含原始脑电信号，而是经过目视检查去除伪影后的干净数据。每个子集包含 100 个信号，每个信号的持续时间为 23.6 秒。子集 A 和 B 分别反映了健康志愿者睁眼和闭眼放松时的脑电活动。子集 C 和 D 侧重于癫痫发作间歇期的大脑活动，子集 C 的数据来自对侧半球的海马体，子集 D 的数据来自致痫区。子集 E 侧重于癫痫发作时的脑电图记录。
% 本文使用5个子集中共500个信号进行分析，每个信号的长度为4000. 图1展示了每个子集的示例信号。

The Bonn Epilepsy Database~\cite{andrzejak2001}, originating from the University of Bonn, consists of EEG data collected from 5 healthy subjects and 5 epilepsy patients. A 128-channel EEG system was used during data collection, following the standard 10-20 electrode placement, with recordings made at a sampling rate of 173.61 Hz. Notably, this dataset does not contain raw EEG signals but rather clean data after visual inspection to remove artifacts. The data are meticulously divided into five subsets labeled A, B, C, D, and E. Each subset contains 100 signals, each with a duration of 23.6 seconds. Subsets A and B reflect the EEG activity of healthy volunteers during relaxation with eyes open and eyes closed, respectively. Subsets C and D focus on interictal brain activity during seizure intervals, with subset C data originating from the hippocampus of the contralateral hemisphere and subset D data originating from the epileptogenic zone. Subset E focuses on EEG recordings during epileptic seizures. A total of 500 signals across 5 subsets were analyzed in this study, with each signal having a length of 4000 samples. Fig.~\ref{fig:eeg_sample} presents example signals from each subset.

\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{eeg_sample.pdf}
\caption{Example EEG signals from the five groups in the dataset.}
\label{fig:eeg_sample}
\end{figure}

% 计算各个组的熵值分布并进行统计分析，结果如图1所示。对于每种熵类型，左侧箱型图展示五个子集的熵值分布，右侧热图显示了每对子集之间的显著性分析结果。统计结果采用Bonferroni 方法进行校正，ns代表没有显著性，*代表p<=0.05,**代表p<=0.005,***代表p<=0.001。

The entropy distributions for each group were computed and statistically analyzed, with the results shown in Fig.~\ref{fig:eeg_entropy}. For each type of entropy, the box plot on the left presents the distribution of entropy values across five subsets, while the heatmap on the right displays the results of significance tests between each pair of subsets.

\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{eeg_entropy.pdf}
\caption{Comparative distributions of entropy values for PE, IPE, AIPE, WPE, RSPE, and ASPE, respectively. Entropy values were computed using the entire data segment of each signal (length of 4097).}
\label{fig:eeg_entropy}
\end{figure}

% 根据复杂度损失理论（complexity loss theory）~cite{costa2005,goldberger2002}以及其他文献对该数据集的结论~cite{xieMultispanTransitionNetworks2024}，预期在不同生理和病理状态下，复杂度会呈现出特定的变化趋势。具体而言，我们假设睁眼状态下的复杂度应高于闭眼状态，正常人群的复杂度应高于癫痫患者，而癫痫患者发作间期的复杂度则应高于发作期。从结果上看，这些熵方法在区分发作期与非发作期方面均表现出一定的有效性。然而，在区分其他状态时，各熵方法的表现则呈现出显著差异。

% PE和RSPE的表现一致，闭眼和发作间期的熵值有些明显的重叠区域。IPE和AIPE在区分闭眼、发作间期和致癫区三个状态时表现不佳，其中IPE甚至无法有效区分睁眼和闭眼状态。值得注意的是，WPE在发作间期和致痫区的熵值竟然小于发作期，这一结论与复杂度损失理论的预期相悖，因此可能是误导性的。
% 相比之下，本文提出的ASPE方法整体表现最佳。在各个生理和病理状态下，ASPE的数据点呈现出更为清晰的分界，表明ASPE在检测幅度变化信号方面具有更稳定的性能。

According to complexity loss theory ~\cite{costa2005,goldberger2002} and findings from other studies on this dataset ~\cite{maDetectionEEGSignals2023}, it is expected that complexity will exhibit specific trends under different physiological and pathological conditions. Specifically, we hypothesized that complexity would be higher in the eyes-open state than in the eyes-closed state, higher in healthy individuals than in epilepsy patients, and higher in epilepsy patients during interictal periods compared to seizure periods. The results show that these entropy methods exhibit a certain degree of effectiveness in distinguishing between seizure and non-seizure periods. However, there are significant differences in their performance when it comes to differentiating other states.

Both PE and RSPE perform similarly, with noticeable overlap in entropy values between the eyes-closed and interictal states. IPE and AIPE perform poorly in distinguishing between the eyes-closed, interictal, and seizure focus states, with IPE even failing to effectively differentiate between eyes-open and eyes-closed conditions. Notably, WPE presents an unexpected result where the entropy values for the interictal and seizure focus states are lower than those during the seizure period, which contradicts the predictions of complexity loss theory and may therefore be misleading. In contrast, the ASPE method proposed in this study demonstrates the best overall performance. The data points for ASPE exhibit clearer separations across different physiological and pathological states, indicating that ASPE provides more stable performance in detecting amplitude-varying signals.

To further validate the effectiveness of these entropy indices in discriminating between different states, classification tasks were performed, and the results are shown in Fig.~\ref{fig:eeg_acc}. For most classification tasks, ASPE demonstrates the best classification performance, followed by RSPE, while PE and WPE exhibit relatively poorer classification performance.

\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{eeg_acc.pdf}
\caption{Accuracy results of the six permutation entropy methods across 7 classification tasks among the eight groups.}
\label{fig:eeg_acc}
\end{figure}


\subsection{ECG arrhythmia dataset}

The ECG signals used in this study come from two datasets: the MIT-BIH Arrhythmia Database (AD)~\cite{moody2001} was used for the patient group, and the MIT-BIH Normal Sinus Rhythm Database (NSRD)~\cite{goldbergerPhysioBankPhysioToolkitPhysioNet2000} served as the control group. The AD was developed by the MIT-BIH Arrhythmia Laboratory and consists of data from 48 subjects, each providing a 30-minute recording of two-channel ambulatory ECG data, sampled at a frequency of 360 Hz. The NSRD was collected by the Beth Israel Deaconess Medical Center and contains data from 18 subjects with normal heart rhythms. This data was also recorded using two-channel ambulatory ECG devices, sampled at 128 Hz, with 24-hour recordings for each subject.

To comprehensively evaluate the performance of the proposed method on data with rich amplitude variation, and inspired by Reference~\cite{martinezcoqDetectionCardiacArrhythmia2022}, we selected raw ECG signals as the primary subject of analysis. The sampling rate of the two datasets was standardized to 128 Hz, and the data were preprocessed using standard methods. Subsequently, for analysis, $20,000$ data points were extracted from each of the two channels for each subject. Examples of 5-second ECG signals from both leads in the AD and NSRD datasets are shown in Fig.~\ref{fig:ecg}~(a).


% 针对两组被试者的每个个体，在其两个通道（EEG1和EEG2）上分别计算了六个排列熵值。为了直观地展示这些熵值之间的关系，我们以EEG1的熵值为横轴，EEG2的熵值为纵轴，绘制了散点图（如图1 (b)所示）。在该图中，圆圈代表95%的置信区间，用于评估数据的离散程度和可靠性。


To quantify the complexity of ECG signals under different heart rate conditions, we computed the entropy values of six permutation entropy methods across the two channels, as shown in Fig.~\ref{fig:ecg}~(c). From the entropy distribution plot, it can be seen that the entropy values of both channels are lower in patients with arrhythmia compared to the normal population, which aligns with the expectations of complexity loss theory. Additionally, except for IPE and AIPE, other permutation entropy measures can effectively discriminate between these two groups.

For each individual subject in both groups, six permutation entropy (PE) values were calculated separately for the two channels (EEG1 and EEG2). To visually present the relationship between these entropy values, we plotted scatter diagrams using the entropy values from EEG1 on the x-axis and those from EEG2 on the y-axis (as shown in Fig.~\ref{fig:ecg}~(b)). In this figure, circles represent the $95\%$ confidence intervals, which are used to assess the degree of data dispersion and reliability.

\begin{figure}[!ht]
\centering
\includegraphics[width=1\linewidth]{ecg.pdf}
\caption{Analysis results of entropy values for six permutation entropy methods in the cardiac arrhythmia dataset. (a) Example ECG signals from two leads for both normal and arrhythmic individuals of ECG1. (b) Example ECG signals from two leads for both normal and arrhythmic individuals of ECG2. (c) Distribution of entropy values for normal individuals and patients with cardiac arrhythmia. Data segments of one hour were extracted from the middle portion of the signal from each channel. Entropy values were computed in one-minute intervals, resulting in a total of 7680 data points, and averaged for each channel. The x-axis represents the entropy values of the ECG1 channel, while the y-axis represents the entropy values of the ECG2 channel. The elliptical area represents the $95\%$ confidence interval.}
\label{fig:ecg}
\end{figure}

% 从熵分布图可以看出，PE和RSPE的散点分布呈现出相似的模式，部分被试者的熵值区域存在明显的重叠，这导致难以有效地区分两组被试者。IPE和AIPE更是如此，丰富的幅度信息对离散化方法带来考验，两组的熵值重叠部分更多，进一步增加了区分的难度。相比之下，WPE的表现略优于PE，两组被试者的散点分布在一定程度上有所区分，但区分度仍然不够显著。然而，本文提出的ASPE方法则展现出了最佳的区分效果。在正常人群中，ASPE的熵值主要集中在右上角的高熵值区域，而在心率失常患者中，ASPE的熵值则主要分布在左下角的低熵值区域。

From the entropy distribution graphs, it can be observed that the scatter distributions of PE and RSPE exhibit similar patterns, with considerable overlap in the entropy regions for some subjects, making it difficult to effectively distinguish between the two groups. This overlap is even more pronounced for IPE and AIPE, as the rich amplitude information challenges the discretization method, resulting in greater overlap between the entropy values of the two groups, further complicating the differentiation. In contrast, WPE performs slightly better than PE, with the scatter distribution of the two groups showing some degree of separation, though the distinction remains insufficient. However, the ASPE method proposed in this paper demonstrates the best discriminatory effect. In the normal group, ASPE entropy values are predominantly concentrated in the upper-right high-entropy region, while in the arrhythmia group, ASPE entropy values are mainly distributed in the lower-left low-entropy region.


% 整体看来，与健康对照组相比，心率失常患者的所有排列熵值均普遍低于正常人群。这一发现与复杂度损失理论相吻合，即疾病状态可能导致系统的动力学复杂性降低，进而表现为熵值的下降。此外，我们的研究结果也与前人的研究相一致，进一步证实了本文方法在分析ECG信号中的潜在价值。
Overall, compared to the healthy control group, all permutation entropy values for the arrhythmia patients are generally lower. This finding aligns with the theory of complexity loss, which suggests that disease states may lead to a reduction in the dynamic complexity of the system, reflected in the decreased entropy values. Moreover, our results are consistent with previous studies~\cite{martinezcoqDetectionCardiacArrhythmia2022,wangMultiscaleIncrementEntropy2022}, further confirming the potential value of the proposed method in analyzing ECG signals.

% 此外，我们还使用两个通道的熵值作为特征进行了分类任务，其平均值与标准差具体结果如图 ~ref{fig:ecg1}~(d-f)所示。比较不同熵指标的分类性能，我们发现 IPE 和 AIPE 的判别能力相对较差，准确率约为 72%$。相比之下，所提出的 ASPE 方法表现优异，在三种情况下的分类准确率都超过了 95 美元/%$，达到了最高水平。这些客观分类结果与之前的主观观察结果一致，都显示了拟议的 ASPE 方法在区分心律失常患者和正常人群方面的卓越能力。
Furthermore, a classification task using entropy values from two channels as features was performed, and the mean and standard deviation results are shown in Table.~\ref{tab:ecg_acc}. Comparing the classification performance of different entropy metrics, we found that IPE and AIPE exhibited relatively poor discriminative ability, with accuracies around $70\%$. In contrast, the proposed ASPE method performed exceptionally well, achieving classification accuracies exceeding $96\%$ in all three cases, reaching the highest level. These objective classification results are consistent with the previous subjective observations, both demonstrating the excellent ability of the proposed ASPE method to discriminate between patients with arrhythmias and the normal population.

\begin{table}[!h]
\centering
\caption{Mean and standard deviation of classification accuracy on different classification tasks.}
\label{tab:ecg_acc}
\begin{ruledtabular}
\begin{tabular}{cccc}
Entropy & ECG1 & ECG2 & ECG1\&ECG2 \\
\hline
PE & $93.55~(\pm0.99\%$)\footnote{The number in brackets represents the standard deviation.} & $90.97~(\pm0.64\%$) & $91.76~(\pm1.13\%$) \\
IPE & 71.18~($\pm$1.58\%) & $70.58~(\pm2.47\%$) & $70.70~(\pm2.07\%$) \\
AIPE & 74.76~($\pm$1.50\%) & $71.24~(\pm1.19\%$) & $74.17~(\pm1.88\%$) \\
WPE & 93.77~($\pm$0.16\%) & $93.84~(\pm0.06\%$) & $93.76~(\pm0.14\%$) \\
RSPE & 93.34~($\pm$0.60\%) & $91.92~(\pm0.33\%$) & $92.28~(\pm0.87\%$) \\
ASPE & \textbf{96.71~$\pm$(0.21\%)}\footnote{The highlighted results indicate the most accurate classification achieved in the current classification task.} & \textbf{96.19~($\pm$0.18\%)} & \textbf{96.74~($\pm$0.17\%)} \\
\end{tabular}
\end{ruledtabular}
\end{table}

%% =========================
\section{Conclusion}
\label{sec:con}

% 生理时间序列数据本身具有非平稳性和动态波动性，表现出各种振幅变化。这些振幅变化反映了生物系统内部复杂的调节机制。准确捕捉和分析这些变化对于了解生物系统的动态以及诊断和治疗潜在疾病至关重要。传统的 PE 是生理时间序列分析中广泛使用的一种方法，它根据顺序关系揭示序列的复杂性，但忽略了隐藏在振幅变化中的其他信息。
% 因此，我们提出了一种新的复杂度测量方法，即振幅敏感熵（ASPE），用于捕捉振幅变化，同时深入挖掘这些变化所包含的复杂度相关信息，从而全面揭示生理时间序列数据的内在特征。现有的振幅敏感方法往往难以深入挖掘振幅变化所包含的复杂信息，尤其是当这种变化伴随着数据幅度的变化时。这就导致熵测量不能准确反映数据的复杂性。
% 相比之下，本文提出的基于变异系数的加权方案综合考虑数据的平均水平和差异成都，为数据的幅度变化提供合理的权重分配标准，提供动力学系统复杂性的全面测量。
Physiological time series data are inherently non-stationary and dynamically fluctuating, exhibiting a variety of amplitude changes. These amplitude changes reflect the complex regulatory mechanisms within biological systems. Accurately capturing and analyzing these changes is crucial for understanding the dynamics of biological systems and for diagnosing and treating potential diseases. Traditional PE, a widely used method in physiological time series analysis, reveals the complexity of sequences based on ordinal relationships but overlooks additional information hidden in amplitude variations. Therefore, we propose a novel complexity measure, termed Amplitude-Sensitive Permutation Entropy (ASPE), to capture amplitude changes while deeply exploring the complexity-related information contained in these variations, thereby comprehensively revealing the intrinsic characteristics of physiological time series data. Existing amplitude-sensitive methods often struggle to delve into the complexity of information contained in amplitude changes, particularly when such changes are accompanied by shifts in data magnitude. This results in entropy measurements that do not accurately reflect the data's complexity. In contrast, the ASPE proposed in this paper, based on the coefficient of variation, comprehensively considers both the average level and the degree of dispersion of the data. It provides a reasonable standard for allocating weights to the amplitude variations of the data, offering a comprehensive measure of the complexity of dynamic systems.

% 模拟数据直观地展示了每种增强策略的概率分布和权重变化，揭示了它们在计算层面的局限性。具体来说，振幅的突然增加和数据幅度的变化会导致离散化结果向极端集聚，从而导致熵测量结果出现明显偏差。同样，加权方法通常会对振幅变化附近的数据赋予极端权重，从而掩盖了其他数据的复杂性信息。这将导致概率分布集中、熵值降低以及缺乏振幅鉴别能力。
% 这些局限性在实际数据应用中也很明显，因为序列中的大量振幅变化会导致较低的判别性能，甚至得出误导性结论。总之，现有方法主要捕捉序列中的 “绝对 ”振幅变化，而 ASPE 则根据变异系数捕捉振幅之间的 “相对 ”变化，更符合排列熵的原理。
% 这使得本文的方法在识别疾病状态和评估其严重程度方面始终优于现有的 PE 方法。
Simulation data intuitively demonstrate the probability distribution and weight changes of each enhancement strategy, revealing their limitations at the computational level. Specifically, sudden increases in amplitude and changes in data magnitude cause discretization results to cluster at the extremes, leading to significant biases in entropy measurements. Similarly, weighted methods often assign extreme weights to data around amplitude changes, masking the complexity information of other data. This results in concentrated probability distributions, reduced entropy, and lack of amplitude discriminability. These limitations are also evident in real data applications, where numerous amplitude changes in the sequence lead to lower discriminative performance and even misleading conclusions. In summary, existing methods primarily capture "absolute" amplitude changes in sequences, whereas ASPE captures "relative" changes between amplitudes based on the coefficient of variation, aligning more closely with the principles of permutation entropy. This renders the method proposed in this paper consistently superior to existing PE methods in identifying disease states and assessing their severity.

% 这项研究的主要贡献在于提供了一个新的视角和更可靠的工具，以加深对生理系统动态的理解。未来的研究可以在这项工作的基础上，探索该方法对其他疾病（如抑郁症和阿尔茨海默病）的适用性，并考虑进行多尺度和多变量分析，以捕捉更复杂的相关性和模式。
The primary contribution of this research lies in offering a new perspective and more reliable tools to deepen the understanding of physiological system dynamics. Future research could expand upon this work by exploring the method's applicability to other diseases, such as depression and Alzheimer's disease, and by considering multi-scale and multivariate analyses to capture more complex correlations and patterns.


\begin{acknowledgments}
This work was supported in part by STI 2030-Major Projects [No.2022ZD0208500, No.2021ZD0202000], in part by the National Key Research and Development Program of China [Grant No.2019YFA0706200], in part by the National Natural Science Foundation of China [Grant No.62372216, Grant No.62102172], in part by Key Program of Natural Science Foundation of Gansu Province [No.22JR5RA410], in part by Gansu Province Science and Technology Program [No. 22JR5RA489], and the Fundamental Research Funds for the Central Universities [No.lzujbky-2022-10].
\end{acknowledgments}

% ===================================

\section*{Data Availability Statement}
The data that support the findings of this study are openly available in https://github.com/Brioal/ASPE.

% ===================================

\section*{Conflict of interest}
The authors declare that they have no conflict of interest.

% ===================================

\bibliography{bibs}

\end{document}
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